Algorithmic Procedures, Generalized Turing Algorithms, and Elementary Recursion Theory

Abstract

Publisher Summary This chapter focuses on algorithmic procedures, generalized turning algorithms, and elementary recursion theory. The theory of recursive or computable functions has been generalized in many ways: (1) to provide techniques for use in other parts of mathematics, particularly logic, set theory, and classical recursive function theory itself, (2) to gain better understanding of advanced recursion theory, for example, degree theory and hierarchy theory, and (3) to gain better understanding of the nature of computation. The generalizations resulting from (1) and (2) have been very successful but have not helped with (3) very much because they usually involve generalizing the notion of finite and the use of infinitely long computations. In classical recursion theory, the natural numbers are computed over, and they not only form the data objects but also can be used for auxiliary arithmetic operations, such as counting. The second section of the paper is a lightning comprehensive survey of how the theorems of elementary recursion theory lift to arbitrary structures when the notion of recursive is taken to mean computable by a gTa.